Application of natural equations to the synthesis of curve generating mechanisms

Abstract In this paper, the coupler curve of the planar four-link mechanism is expressed using the natural equation in the form of a single-value function, κ = g(s). Here κ is the curvature of the coupler curve and s is the arc length. The natural equation of the desired curve κ = f(s) is given by means of the spline function. The condition is made clear that an open planar curve expressed using the natural equation becomes closed, which is cut in three portions and connected with two circular arcs. A new method for synthesizing the curve generating mechanism is presented based on the one-to-one correspondence between the points on the coupler curve and the ones on the desired curve so that the norm with respect to their natural equations may be minimized and the coordinate transformation of the coupler curve so that the maximum difference between both curves may be minimized. Numerical examples included here show that the method has attractive convergence properties in a search domain composed of kinematic constants.